Shimming methods are known in the art and allow the correction of magnetic field inhomogeneity in magnetic resonance imaging machines before imaging.
At the beginning, the magnetic field generated by the magnet is strongly affected by inhomogeneity caused by manufacturing tolerances and environmental interferences, which have highly detrimental effects on image quality.
The magnet is preferably a permanent magnet, although it can also be an electromagnet, namely, a superconducting electromagnet.
In the so called passive shimming, correction elements are placed on the surfaces of the pole pieces of the magnet to influence the magnetic field and bring it to an adequate homogeneity level for magnetic resonance imaging.
The grid for positioning correction elements as defined hereinbefore is intended to be formed on a ferromagnetic plate, commonly known as pole piece, which is placed orthogonal to the main field direction.
Therefore, the grid is placed on the magnet, the term magnet being generally intended not only as the macroscopic distribution for generating the main magnetic field, but also as any other ferromagnetic part that forms the system and hence also the pole piece.
The magnitude parameters of the correction elements as defined above include the geometric dimensions of the element and the magnetic properties of the material that forms the element.
In case of permanent magnets, the correction elements that are used for passive shimming are blocks of magnetized material.
Concerning the determination of position and magnitude parameters of one or more correction elements for obtaining the desired field characteristics, theoretical equations are provided in the art, which define the magnetic field generated in one point in space by a magnetic dipole located in one point of a positioning grid placed on a ferromagnetic pole piece having an infinite magnetic permeability.
Therefore, if a single correction element is defined, then all the expansions of the field generated at the points of the sampling grid may be calculated, by moving the single element from time to time over the positioning grid.
The expansions so created may be mathematically grouped into a so-called “effect matrix”.
Using this effect matrix, appropriate mathematical minimization and/or optimization algorithms, such as pseudo-inversion, quadratic programming, Tikhonov regularization or else may allow determination of the best distribution of correction elements, in terms of their position on the positioning grid and magnitude parameters, to minimize or even eliminate field inhomogeneity.
The magnetic field shimming process is known to be mathematically based on the exact solution to Laplace's equation using the technique of separation of variables.
In short, the determination of an appropriate coordinate transformation from a Cartesian space to a “curvilinear” space allows determination of an exact solution to Laplace's equation, consisting of the product of three functions of one variable, which means that:
considering F(ξ, λ, φ) as a solution to the Equation ∇2F(ξ, λ, φ)=0 the following may be obtained:F(ξ,λ,φ)=Π(ξ)*Λ(λ)*Φ(φ)
As is further known, Laplace's equation is a differential equation and admits infinite solutions. One of them is chosen, which fulfills the boundary conditions defined on a surface D.
In this type of “boundary value problems,” the selected surface D is usually (but not necessarily) a “coordinate” surface of the “curvilinear” space, i.e the geometric place in that defined “curvilinear” space where one of the three variables is constant. For example, in the case of spherical curvilinear coordinates (r, θ, φ), the coordinate surface under consideration is that with r=R=constant.
It should be noted that, for all “curvilinear” spaces in which the method of separation of variables can be used, the selected coordinate transformation causes one of the coordinate surfaces to be a sphere. In other words, and using an explanatory example, if we consider an ellipsoidal curvilinear space, the determined coordinate transformation (which allows separation of variables in the Laplace's equation) will cause an ellipsoid in the Cartesian space to become a spherical surface in the ellipsoidal space (which means that a “radius” of the ellipsoid may be determined).
There are possible cases of curvilinear spaces, in which such method may be used analytically, such that an exact solution to Laplace's equation may be obtained.
In this case, the associated surfaces are known as “quadrics” because the maximum degree of the equations that represent them is 2, and are closed surfaces or, in mathematical terms, are simply connected surfaces, e.g. a cube, a sphere, an oblate spheroid and an ellipsoid.
Finally, a linear superposition, i.e. a series of solution functions for Laplace's equation of this type will also be, due to the linearity of the Laplacian operator, a solution to Laplace's equation.
Bearing the above in mind, it shall be noted that in dedicated MRI, the image volume is usually defined as a cube or a parallelepiped within the cavity of the system and the hypothesis that the shimming volume contains the entire field of view (FOV), i.e. the image volume, is not always completely fulfilled.
For example, if the FOV is a cube with a side length of 140 mm, as is typical in magnetic resonance imaging machines, and the shimming surface is a quadric, and particularly an oblate spheroid with axes of 82×95 mm, the shimming volume only partially contains the image volume.
This involves the detrimental effect that shimming, i.e. homogeneity correction of the magnetic field, may be performed in areas outside the FOV, and that some areas of the latter will remain with a non-homogeneous magnetic field.
While this condition may be acceptable in certain cases, when allowed by the anatomy of the patient under examination, in other cases it leads to unacceptable artifacts, such as in sagittal T2-weighted Fast Spin Echo (FSE).
The attempt to contain the entire image volume with one of the quadrics inevitably leads to a considerable increase of the geometric dimensions of the apparatus.
This will necessarily increase the size, weight, amount of magnetic material, etc., thereby leading to a rapid increase of the costs of manufacture, installation, maintenance and operation of the magnetic resonance imaging machine.
One of the quadrics is the cube, and one may wonder why the most logical solution to entirely contain the FOV, a cubic shimming volume, is not chosen.
The answer to this is not found in mathematics, but in physics: the static shimming process requires a convergent solution series for Laplace's equation, i.e. with a finite limit of partial sums, and the problem must also be simplified by dividing it into independent or orthogonal subproblems or symmetries.
This possibility is prevented in the Cartesian-cubic case, if boundary conditions are other than zero.
This is clearly the case of MRI systems, in which boundary conditions are given on an obviously non-zero magnetic field B, both for field value conditions known as “Dirichlet” and for conditions on the value of the field derivative in a spatial direction, known as “Neumann”.
Therefore, there currently exists a geometric problem preventing shimming volume-FOV overlap, which is yet unsolved by prior art shimming methods.